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Properties

Label 10-1531e5-1531.1530-c0e5-0-0
Degree $10$
Conductor $8.412\times 10^{15}$
Sign $1$
Analytic cond. $0.260412$
Root an. cond. $0.874110$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·4-s − 5-s − 7-s + 5·9-s − 11-s − 13-s + 15·16-s − 17-s − 5·20-s − 5·28-s − 29-s − 31-s + 35-s + 25·36-s − 43-s − 5·44-s − 5·45-s − 5·52-s − 53-s + 55-s − 59-s − 5·63-s + 35·64-s + 65-s − 67-s − 5·68-s − 73-s + ⋯
L(s)  = 1  + 5·4-s − 5-s − 7-s + 5·9-s − 11-s − 13-s + 15·16-s − 17-s − 5·20-s − 5·28-s − 29-s − 31-s + 35-s + 25·36-s − 43-s − 5·44-s − 5·45-s − 5·52-s − 53-s + 55-s − 59-s − 5·63-s + 35·64-s + 65-s − 67-s − 5·68-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1531^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(1531^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(1531^{5}\)
Sign: $1$
Analytic conductor: \(0.260412\)
Root analytic conductor: \(0.874110\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1531} (1530, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 1531^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.032598259\)
\(L(\frac12)\) \(\approx\) \(5.032598259\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad1531$C_1$ \( ( 1 - T )^{5} \)
good2$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
5$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
7$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
11$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
13$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
17$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
29$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
31$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
43$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
53$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
59$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
67$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
73$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
79$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
89$C_{10}$ \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{5}( 1 + T )^{5} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.97040023623706149889386684509, −5.80150507369096583169651274970, −5.68838739796883029242303975700, −5.49177739735468433016380132065, −5.33736606180943247480527429751, −4.92051431137945220000256668262, −4.74351554297343509019658520803, −4.64377981036241726533116032273, −4.55204980334237853842083101743, −4.06220249023891730864035423745, −3.87537909717498649352209858430, −3.82841417435357362791309364464, −3.59854711966353092494489129247, −3.48834068356195125042800746504, −3.30531412377133581613473240663, −2.84385906641385212524897598364, −2.76432988515172115980272498358, −2.50435507747885799783187472486, −2.44163849471464711907605015809, −2.14853199560932274427792789252, −1.74356079335354773109385479993, −1.65079566063042676137057368625, −1.61240373700180327870933923135, −1.24334905755191109794358118034, −1.19243923608767810803720805359, 1.19243923608767810803720805359, 1.24334905755191109794358118034, 1.61240373700180327870933923135, 1.65079566063042676137057368625, 1.74356079335354773109385479993, 2.14853199560932274427792789252, 2.44163849471464711907605015809, 2.50435507747885799783187472486, 2.76432988515172115980272498358, 2.84385906641385212524897598364, 3.30531412377133581613473240663, 3.48834068356195125042800746504, 3.59854711966353092494489129247, 3.82841417435357362791309364464, 3.87537909717498649352209858430, 4.06220249023891730864035423745, 4.55204980334237853842083101743, 4.64377981036241726533116032273, 4.74351554297343509019658520803, 4.92051431137945220000256668262, 5.33736606180943247480527429751, 5.49177739735468433016380132065, 5.68838739796883029242303975700, 5.80150507369096583169651274970, 5.97040023623706149889386684509

Graph of the $Z$-function along the critical line